L(s) = 1 | + (1.37 + 0.320i)2-s + (1.79 + 0.883i)4-s + (−0.707 + 0.707i)5-s − 4.02i·7-s + (2.18 + 1.79i)8-s + (−1.20 + 0.747i)10-s + (0.646 − 0.646i)11-s + (4.91 + 4.91i)13-s + (1.29 − 5.54i)14-s + (2.43 + 3.17i)16-s + 2.70·17-s + (−0.438 − 0.438i)19-s + (−1.89 + 0.643i)20-s + (1.09 − 0.683i)22-s − 3.60i·23-s + ⋯ |
L(s) = 1 | + (0.973 + 0.226i)2-s + (0.897 + 0.441i)4-s + (−0.316 + 0.316i)5-s − 1.52i·7-s + (0.773 + 0.633i)8-s + (−0.379 + 0.236i)10-s + (0.195 − 0.195i)11-s + (1.36 + 1.36i)13-s + (0.345 − 1.48i)14-s + (0.609 + 0.792i)16-s + 0.656·17-s + (−0.100 − 0.100i)19-s + (−0.423 + 0.143i)20-s + (0.234 − 0.145i)22-s − 0.750i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.82821 + 0.373767i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.82821 + 0.373767i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 - 0.320i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + 4.02iT - 7T^{2} \) |
| 11 | \( 1 + (-0.646 + 0.646i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.91 - 4.91i)T + 13iT^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 19 | \( 1 + (0.438 + 0.438i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.60iT - 23T^{2} \) |
| 29 | \( 1 + (2.00 + 2.00i)T + 29iT^{2} \) |
| 31 | \( 1 - 4.30T + 31T^{2} \) |
| 37 | \( 1 + (0.743 - 0.743i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.603iT - 41T^{2} \) |
| 43 | \( 1 + (5.03 - 5.03i)T - 43iT^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + (4.07 - 4.07i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.22 - 1.22i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.98 + 6.98i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.24 - 5.24i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.7iT - 71T^{2} \) |
| 73 | \( 1 + 1.30iT - 73T^{2} \) |
| 79 | \( 1 - 0.611T + 79T^{2} \) |
| 83 | \( 1 + (1.29 + 1.29i)T + 83iT^{2} \) |
| 89 | \( 1 - 10.9iT - 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.81883948143314630762214521210, −9.790835352612945957709325954562, −8.413924591464378004932172815006, −7.64999965942335164187155871083, −6.66096245746064315027667918604, −6.28571306747345957794917779839, −4.70761610436094261944398669158, −4.00950791964188761816271883101, −3.24146669520741305498105963315, −1.47272888515014550552959821402,
1.50078533201252992084416585312, 2.92810919078422211023055176748, 3.71440378444815378298657605830, 5.14239469183125758439583552809, 5.64945834467025837379876277576, 6.50098787070444205388754348356, 7.84527591876922130269085970682, 8.546235152004999723128369705532, 9.621304011895306485145459225956, 10.58068698766768333017343784183